p t (t) 







P a «P* 



-I ♦! 



at 



Figure. 2.. .The sound pressure radiated by a bubble in response to an environmental pressure given by 

 Pelt) =: P — P a sech'at, which is similar to the pressure on a bubble moving around the surface 

 of a cylinder in a flow. The curves indicate the development of non-linearity for successively 

 smaller values of P relative to P a . The solid curve represents the linear range. 



adiabatic gas law, P — P g (R /R) s r — (2T/R), where R is the equilibrium radius of 

 the bubble at the mean gas pressure P g — P + 2T/R ; and T is the surface tension.* 

 The non-linear effects can be investigated most easily if the environmental pres- 

 sure is assumed to vary in some definite way. For comparison with the linear theory, 

 consider the fluctuation given by p e (t) — P — P a sech 2 at, which results in sound with 

 a waveform like that of Eq. (11) if P a <^.P . The changes in the waveform associated 

 with larger values of P a are illustrated qualitatively in Fig. 2. For P a <^.P , Eq. (12) 

 becomes linear and the waveform of the radiated pressure (solid curve) is just that 

 given by Eq. (11). As P a is made larger, non-linearity occurs first in the term (R /R) z 

 in Eq.(12), and the negative peak in the sound pressure grows rapidly (dashed curve). 



* All pressures are measured relative to the vapor pressure of the water, in order to 

 avoid a repetitive constant. 



246 



